On the Closed-form Weight Enumeration of Polar Codes: 1.5$d$-weight Codewords
The weight distribution of error correction codes is a critical determinant of their error-correcting performance, making enumeration of utmost importance. In the case of polar codes, the minimum weight $\wm$ (which is equal to minimum distance $d$) is the only weight for which an explicit enumerato...
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| creator | Rowshan, Mohammad Drăgoi, Vlad-Florin Yuan, Jinhong |
| description | The weight distribution of error correction codes is a critical determinant
of their error-correcting performance, making enumeration of utmost importance.
In the case of polar codes, the minimum weight $\wm$ (which is equal to minimum
distance $d$) is the only weight for which an explicit enumerator formula is
currently available. Having closed-form weight enumerators for polar codewords
with weights greater than the minimum weight not only simplifies the
enumeration process but also provides valuable insights towards constructing
better polar-like codes. In this paper, we contribute towards understanding the
algebraic structure underlying higher weights by analyzing Minkowski sums of
orbits. Our approach builds upon the lower triangular affine (LTA) group of
decreasing monomial codes. Specifically, we propose a closed-form expression
for the enumeration of codewords with weight $1.5\wm$. Our simulations
demonstrate the potential for extending this method to higher weights. |
| doi_str_mv | 10.48550/arxiv.2305.02921 |
| format | Article |
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of their error-correcting performance, making enumeration of utmost importance.
In the case of polar codes, the minimum weight $\wm$ (which is equal to minimum
distance $d$) is the only weight for which an explicit enumerator formula is
currently available. Having closed-form weight enumerators for polar codewords
with weights greater than the minimum weight not only simplifies the
enumeration process but also provides valuable insights towards constructing
better polar-like codes. In this paper, we contribute towards understanding the
algebraic structure underlying higher weights by analyzing Minkowski sums of
orbits. Our approach builds upon the lower triangular affine (LTA) group of
decreasing monomial codes. Specifically, we propose a closed-form expression
for the enumeration of codewords with weight $1.5\wm$. Our simulations
demonstrate the potential for extending this method to higher weights.</description><identifier>DOI: 10.48550/arxiv.2305.02921</identifier><language>eng</language><subject>Computer Science - Information Theory ; Mathematics - Information Theory</subject><creationdate>2023-05</creationdate><rights>http://creativecommons.org/licenses/by/4.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,782,887</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/2305.02921$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.2305.02921$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Rowshan, Mohammad</creatorcontrib><creatorcontrib>Drăgoi, Vlad-Florin</creatorcontrib><creatorcontrib>Yuan, Jinhong</creatorcontrib><title>On the Closed-form Weight Enumeration of Polar Codes: 1.5$d$-weight Codewords</title><description>The weight distribution of error correction codes is a critical determinant
of their error-correcting performance, making enumeration of utmost importance.
In the case of polar codes, the minimum weight $\wm$ (which is equal to minimum
distance $d$) is the only weight for which an explicit enumerator formula is
currently available. Having closed-form weight enumerators for polar codewords
with weights greater than the minimum weight not only simplifies the
enumeration process but also provides valuable insights towards constructing
better polar-like codes. In this paper, we contribute towards understanding the
algebraic structure underlying higher weights by analyzing Minkowski sums of
orbits. Our approach builds upon the lower triangular affine (LTA) group of
decreasing monomial codes. Specifically, we propose a closed-form expression
for the enumeration of codewords with weight $1.5\wm$. Our simulations
demonstrate the potential for extending this method to higher weights.</description><subject>Computer Science - Information Theory</subject><subject>Mathematics - Information Theory</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2023</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNotjztPwzAUhb10qAo_oBMeujrcOLYTd0NReUhFZajEGN3E1zRSEiMnUPj30Md0pHM-HeljbJlCogqt4R7jT_udyAx0AtLKdM5edwOfDsTLLozkhA-x5-_Ufhwmvhm-eoo4tWHgwfO30GHkZXA0rnma6JVbieOFPJXHEN14w2Yeu5Fur7lg-8fNvnwW293TS_mwFWjyVOTKKSqkUmAaACTnbZ1r8oXMkcDXWur_yRRI2pgMrWoUKKt8DU1jjTPZgt1dbs8-1Wdse4y_1cmrOntlfwSORoA</recordid><startdate>20230504</startdate><enddate>20230504</enddate><creator>Rowshan, Mohammad</creator><creator>Drăgoi, Vlad-Florin</creator><creator>Yuan, Jinhong</creator><scope>AKY</scope><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20230504</creationdate><title>On the Closed-form Weight Enumeration of Polar Codes: 1.5$d$-weight Codewords</title><author>Rowshan, Mohammad ; Drăgoi, Vlad-Florin ; Yuan, Jinhong</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a671-74d4e824406c00aedf9b75ef827ae0fb52540668ae5663a94c40494fb0cc96d63</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2023</creationdate><topic>Computer Science - Information Theory</topic><topic>Mathematics - Information Theory</topic><toplevel>online_resources</toplevel><creatorcontrib>Rowshan, Mohammad</creatorcontrib><creatorcontrib>Drăgoi, Vlad-Florin</creatorcontrib><creatorcontrib>Yuan, Jinhong</creatorcontrib><collection>arXiv Computer Science</collection><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Rowshan, Mohammad</au><au>Drăgoi, Vlad-Florin</au><au>Yuan, Jinhong</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On the Closed-form Weight Enumeration of Polar Codes: 1.5$d$-weight Codewords</atitle><date>2023-05-04</date><risdate>2023</risdate><abstract>The weight distribution of error correction codes is a critical determinant
of their error-correcting performance, making enumeration of utmost importance.
In the case of polar codes, the minimum weight $\wm$ (which is equal to minimum
distance $d$) is the only weight for which an explicit enumerator formula is
currently available. Having closed-form weight enumerators for polar codewords
with weights greater than the minimum weight not only simplifies the
enumeration process but also provides valuable insights towards constructing
better polar-like codes. In this paper, we contribute towards understanding the
algebraic structure underlying higher weights by analyzing Minkowski sums of
orbits. Our approach builds upon the lower triangular affine (LTA) group of
decreasing monomial codes. Specifically, we propose a closed-form expression
for the enumeration of codewords with weight $1.5\wm$. Our simulations
demonstrate the potential for extending this method to higher weights.</abstract><doi>10.48550/arxiv.2305.02921</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Computer Science - Information Theory Mathematics - Information Theory |
| title | On the Closed-form Weight Enumeration of Polar Codes: 1.5$d$-weight Codewords |
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